Discrete Math For Computing II

Transcription

1 Discrete Math For Computing II Main Text: Topics in enumeration; principle of inclusion and exclusion, Partial orders and lattices. Algorithmic complexity; recurrence relations, Graph theory. Prerequisite: CS 2305 "Discrete Mathematics and its Applications" Kenneth Rosen, 5th Edition, McGraw Hill.

2 Contact Information B. Prabhakaran Department of Computer Science University of Texas at Dallas Mail Station EC 31, PO Box Richardson, TX Phone: ; Fax: URL: Office: ES Office Hours: pm & pm. Tuesdays; pm Thursdays Other times by appointments through Announcements: Made in class and on course web page. TA: TBA.

3 Course Outline Selected topics in chapters 6 through 9. Chapter 6: Advanced Counting Techniques: recurrence relations, principle of inclusion and exclusion Chapter 7: Relations: properties of binary relations, representing relations, equivalence relations, partial orders Chapter 8: Graphs: graph representation, isomorphism, Euler paths, shortest path algorithms, planar graphs, graph coloring Chapter 9: Trees: tree applications, tree traversal, trees and sorting, spanning trees

4 Course ABET Objectives Ability to construct and solve recurrence relations Ability to use the principle of inclusion and exclusion to solve problems Ability to understand binary relations and their applications Ability to recognize and use equivalence relations and partial orderings Ability to use and construct graphs and graph terminology Ability to apply the graph theory concepts of Euler and Hamilton circuits Ability to identify and use planar graphs and shortest path problems Ability to use and construct trees and tree terminology Ability to use and construct binary search trees

5 Evaluation 2 Mid-terms: in class. 75 minutes. Mix of MCQs (Multiple Choice Questions) & Short Questions. 1 Final Exam: 75 minutes or 2 hours (depending on class room availability). Mix of MCQs and Short Questions. 2-3 Quizzes: 5-6 MCQs or very short questions. 15 minutes each. Homeworks/assignments: 3 or 4 spread over the semester.

6 Homeworks Each homework will be for 10 marks. Homeworks Submission: Submit on paper to TA/Instructor.

7 Grading Home works: 5% Quizzes: 10% Mid-terms : 50% Final: 35%

8 Likely Letter Grades A-, A, A+: % B-, B, B+: 80-90% C-, C, C+: 70-80% D-, D, D+: 60-70% F: Below 60% Class average will influence above the ranges.

9 Schedule Quizzes: Dates announced in class & web, a week ahead. Mostly just before midterms and final. Mid-term 1 : February 10, 2005 Mid-term 2: March 24, 2005 Final Exam: 11am, April 26, 2005 (As per UTD schedule) Subject to minor changes Quiz and homework schedules will be announced in class and course web page, giving sufficient time for preparation.

10 Cheating Academic dishonesty will be taken seriously. Cheating students will be handed over to Head/Dean for further action. Remember: home works (and exams too!) are to be done individually. Any kind of cheating in home works/exams will be dealt with as per UTD guidelines.

11 Spring Break! Conference Travel in the week starting March 1 st. Propose a 2-hour make up class (with 15 minute break between the hours) on a Monday morning?

12 Graph Theory Chapter 8

13 Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities Scheduling exams and assign channels to television stations

14 Topics Covered Definitions Types Terminology Representation Sub-graphs Connectivity Hamilton and Euler definitions Shortest Path Planar Graphs Graph Coloring

15 Definitions - Graph A generalization of the simple concept of a set of dots, links, edges or arcs. Representation: Graph G =(V, E) consists set of vertices denoted by V, or by V(G) and set of edges E, or E(G)

16 Definitions Edge Type Directed: Ordered pair of vertices. Represented as (u, v) directed from vertex u to v. u v Undirected: Unordered pair of vertices. Represented as {u, v}. Disregards any sense of direction and treats both end vertices interchangeably. u v

17 Definitions Edge Type Loop: A loop is an edge whose endpoints are equal i.e., an edge joining a vertex to it self is called a loop. Represented as {u, u} = {u} u Multiple Edges: Two or more edges joining the same pair of vertices.

18 Definitions Graph Type Simple (Undirected) Graph: consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges (undirected) Representation Example: G(V, E), V = {u, v, w}, E = {{u, v}, {v, w}, {u, w}} u v w

19 Definitions Graph Type Multigraph: G(V,E), consists of set of vertices V, set of Edges E and a function f from E to {{u, v} u, v V, u v}. The edges e1 and e2 are called multiple or parallel edges if f (e1) = f (e2). Representation Example: V = {u, v, w}, E = {e 1, e 2, e 3 } u e 1 e 2 w v e 3

20 Definitions Graph Type Pseudograph: G(V,E), consists of set of vertices V, set of Edges E and a function F from E to {{u, v} u, v Î V}. Loops allowed in such a graph. Representation Example: V = {u, v, w}, E = {e 1, e 2, e 3, e 4 } u e 1 w e 4 e 2 v e 3

21 Definitions Graph Type Directed Graph: G(V, E), set of vertices V, and set of Edges E, that are ordered pair of elements of V (directed edges) Representation Example: G(V, E), V = {u, v, w}, E = {(u, v), (v, w), (w, u)} u v w

22 Definitions Graph Type Directed Multigraph: G(V,E), consists of set of vertices V, set of Edges E and a function f from E to {{u, v} u, v V}. The edges e1 and e2 are multiple edges if f(e1) = f(e2) Representation Example: V = {u, v, w}, E = {e 1, e 2, e 3, e 4 } u e 1 e 2 u e 4 u e 3

23 Definitions Graph Type Type Edges Multiple Edges Allowed? Loops Allowed? Simple Graph undirected No No Multigraph undirected Yes No Pseudograph undirected Yes Yes Directed Graph directed No Yes Directed Multigraph directed Yes Yes

24 Terminology Undirected graphs u and v are adjacent if {u, v} is an edge, e is called incident with u and v. u and v are called endpoints of {u, v} Degree of Vertex (deg (v)): the number of edges incident on a vertex. A loop contributes twice to the degree (why?). Pendant Vertex: deg (v) =1 Isolated Vertex: deg (v) = 0 Representation Example: For V = {u, v, w}, E = { {u, w}, {u, w}, (u, v) }, deg (u) = 2, deg (v) = 1, deg (w) = 1, deg (k) = 0, w and v are pendant, k is isolated u v k w

25 Terminology Directed graphs For the edge (u, v), u is adjacent to v OR v is adjacent from u, u Initial vertex, v Terminal vertex In-degree (deg - (u)): number of edges for which u is terminal vertex Out-degree (deg + (u)): number of edges for which u is initial vertex Note: A loop contributes 1 to both in-degree and out-degree (why?) Representation Example: For V = {u, v, w}, E = { (u, w), ( v, w), (u, v) }, deg - (u) = 0, deg + (u) = 2, deg - (v) = 1, deg + (v) = 1, and deg - (w) = 2, deg + (u) = 0 u v w

26 Theorems: Undirected Graphs Theorem 1 The Handshaking theorem: 2 e = v V v (why?) Every edge connects 2 vertices

27 Theorems: Undirected Graphs Theorem 2: An undirected graph has even number of vertices with odd degree Pr oof V1is the set of even degree vertices and V2 refers to odd degree vertices 2e = v V deg(v) = deg (v) is even for v V second term u V v V deg(u) + v V Hence second term is also even deg(v) = even 2 1 the last inequality is even since sum is 2e. 1, The sum of the last two terms on the right hand side of 2 deg(v) The first term in the right hand side of the last inequality is even.

28 Theorems: directed Graphs Theorem 3: deg + (u) = deg - (u) = E

29 Simple graphs special cases Complete graph: K n, is the simple graph that contains exactly one edge between each pair of distinct vertices. Representation Example: K 1, K 2, K 3, K 4 K 1 K 2 K 3 K 4

30 Simple graphs special cases Cycle: C n, n 3 consists of n vertices v 1, v 2, v 3 v n and edges {v 1, v 2 }, {v 2, v 3 }, {v 3, v 4 } {v n-1, v n }, {v n, v 1 } Representation Example: C 3, C 4 C 3 C 4

31 Simple graphs special cases Wheels: W n, obtained by adding additional vertex to Cn and connecting all vertices to this new vertex by new edges. Representation Example: W 3, W 4 W 3 W 4

32 Simple graphs special cases N-cubes: Q n, vertices represented by 2n bit strings of length n. Two vertices are adjacent if and only if the bit strings that they represent differ by exactly one bit positions Representation Example: Q 1, Q Q 1 Q 2

33 Bipartite graphs In a simple graph G, if V can be partitioned into two disjoint sets V 1 and V 2 such that every edge in the graph connects a vertex in V 1 and a vertex V 2 (so that no edge in G connects either two vertices in V 1 or two vertices in V 2 ) Application example: Representing Relations Representation example: V 1 = {v 1, v 2, v 3 } and V 2 = {v 4, v 5, v 6 }, v 1 v 4 v 2 v 5 v 3 v 6 V 1 V 2

34 Complete Bipartite graphs K m,n is the graph that has its vertex set portioned into two subsets of m and n vertices, respectively There is an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset. Representation example: K 2,3, K 3,3 K 2,3 K 3,3

35 Subgraphs A subgraph of a graph G = (V, E) is a graph H =(V, E ) where V is a subset of V and E is a subset of E Application example: solving sub-problems within a graph Representation example: V = {u, v, w}, E = ({u, v}, {v, w}, {w, u}}, H 1, H 2 u u u v w v w v G H 1 H 2

36 Subgraphs G = G1 U G2 wherein E = E1 U E2 and V = V1 U V2, G, G1 and G2 are simple graphs of G Representation example: V1 = {u, w}, E1 = {{u, w}}, V2 = {w, v}, E1 = {{w, v}}, V = {u, v,w}, E = {{{u, w}, {{w, v}} u u w w v w v G1 G2 G

37 Representation Incidence (Matrix): Most useful when information about edges is more desirable than information about vertices. Adjacency (Matrix/List): Most useful when information about the vertices is more desirable than information about the edges. These two representations are also most popular since information about the vertices is often more desirable than edges in most applications

38 G Representation- Incidence Matrix = (V, E) be an unditected graph. Suppose that v 1, v 2, v 3,, v n are the vertices and e 1, e 2,, e m are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the nx m matrix M = [m ij ], where m ij = 1 0 when edge e otherwise j is incident with v i Can also be used to represent : Multiple edges: by using columns with identical entries, since these edges are incident with the same pair of vertices Loops: by using a column with exactly one entry equal to 1, corresponding to the vertex that is incident with the loop

39 Representation- Incidence Matrix Representation Example: G = (V, E) e1 u e2 v u e e e v e3 w w 0 1 1

40 There For This Representation- Adjacency Matrix is an N x N matrix, where V = N, the Adjacenct Matrix (NxN) A = [a ij ] For undirected graph a ij 1 = 0 if {v i, v j}is an edge of otherwise G directed graph a ij 1 if (v i, v j) is an edge of = 0 otherwise G makes it easier to find subgraphs, and to reverse graphs if needed.

41 Representation- Adjacency Matrix Adjacency is chosen on the ordering of vertices. Hence, there as are as many as n! such matrices. The adjacency matrix of simple graphs are symmetric (a ij = a ji ) (why?) When there are relatively few edges in the graph the adjacency matrix is a sparse matrix Directed Multigraphs can be represented by using aij = number of edges from v i to v j

42 Representation- Adjacency Matrix Example: Undirected Graph G (V, E) u v u w v v w u w 1 1 0

43 Representation- Adjacency Matrix Example: directed Graph G (V, E) u v u w v v w u w 1 0 0

44 Representation- Adjacency List Each node (vertex) has a list of which nodes (vertex) it is adjacent Example: undirectd graph G (V, E) u node u Adjacency List v, w v w v w, u w u, v

45 Graph - Isomorphism G1 = (V1, E2) and G2 = (V2, E2) are isomorphic if: There is a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f (a) and f (b) are adjacent in G2, for all a and b in V1. Function f is called isomorphism Application Example: In chemistry, to find if two compounds have the same structure

46 Graph - Isomorphism Representation example: G1 = (V1, E1), G2 = (V2, E2) f(u 1 ) = v 1, f(u 2 ) = v 4, f(u 3 ) = v 3, f(u 4 ) = v 2, u 1 u 2 v 1 v 2 u 4 u 3 v 4 v 3

47 Connectivity Basic Idea: In a Graph Reachability among vertices by traversing the edges Application Example: - In a city to city road-network, if one city can be reached from another city. - Problems if determining whether a message can be sent between two computer using intermediate links - Efficiently planning routes for data delivery in the Internet

48 Connectivity Path A Path is a sequence of edges that begins at a vertex of a graph and travels along edges of the graph, always connecting pairs of adjacent vertices. Representation example: G = (V, E), Path P represented, from u to v is {{u, 1}, {1, 4}, {4, 5}, {5, v}} u v 4 5

49 Connectivity Path Definition for Directed Graphs A Path of length n (> 0) from u to v in G is a sequence of n edges e 1, e 2, e 3,, e n of G such that f (e 1 ) = (x o, x 1 ), f (e 2 ) = (x 1, x 2),, f (e n ) = (x n-1, x n), where x 0 = u and x n = v. A path is said to pass through x 0, x 1,, x n or traverse e 1, e 2, e 3,, e n For Simple Graphs, sequence is x 0, x 1,, x n In directed multigraphs when it is not necessary to distinguish between their edges, we can use sequence of vertices to represent the path Circuit/Cycle: u = v, length of path > 0 Simple Path: does not contain an edge more than once

50 Connectivity Connectedness Undirected Graph An undirected graph is connected if there exists is a simple path between every pair of vertices Representation Example: G (V, E) is connected since for V = {v 1, v 2, v 3, v 4, v 5 }, there exists a path between {v i, v j }, 1 i, j 5 v 1 v 3 v 4 v 2 v 5

51 Connectivity Connectedness Undirected Graph Articulation Point (Cut vertex): removal of a vertex produces a subgraph with more connected components than in the original graph. The removal of a cut vertex from a connected graph produces a graph that is not connected Cut Edge: An edge whose removal produces a subgraph with more connected components than in the original graph. Representation example: G (V, E), v 3 is the articulation point or edge {v 2, v 3 }, the number of connected components is 2 (> 1) v 1 v 3 v 5 v 2 v 4

52 Connectivity Connectedness Directed Graph A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph A directed graph is weakly connected if there is a (undirected) path between every two vertices in the underlying undirected path A strongly connected Graph can be weakly connected but the vice-versa is not true (why?)

53 Connectivity Connectedness Directed Graph Representation example: G1 (Strong component), G2 (Weak Component), G3 is undirected graph representation of G2 or G1 G1 G2 G3

54 Connectivity Connectedness Directed Graph Strongly connected Components: subgraphs of a Graph G that are strongly connected Representation example: G1 is the strongly connected component in G G G1

55 Isomorphism - revisited A isomorphic invariant for simple graphs is the existence of a simple circuit of length k, k is an integer > 2 (why?) Representation example: G1 and G2 are isomorphic since we have the invariants, similarity in degree of nodes, number of edges, length of circuits G1 G2

56 Counting Paths Theorem: Let G be a graph with adjacency matrix A with respect to the ordering v 1, v2,, V n (with directed on undirected edges, with multiple edges and loops allowed). The number of different paths of length r from Vi to Vj, where r is a positive integer, equals the (i, j) th entry of (adjacency matrix) A r. Proof: By Mathematical Induction. Base Case: For the case N = 1, a ij =1 implies that there is a path of length 1. This is true since this corresponds to an edge between two vertices. We assume that theorem is true for N = r and prove the same for N = r +1. Assume that the (i, j) th entry of A r is the number of different paths of length r from v i to v j. By induction hypothesis, b ik is the number of paths of length r from v i to v k.

57 Counting Paths Case r +1: In A r+1 = A r. A, The (i, j) th entry in A r+1, b i1 a 1j + b i2 a 2j + + b in a nj where b ik is the (i, j) th entry of A r. By induction hypothesis, b ik is the number of paths of length r from v i to v k. The (i, j) th entry in A r+1 corresponds to the length between i and j and the length is r+1. This path is made up of length r from v i to v k and of length from v k to vj. By product rule for counting, the number of such paths is b ik* a kj The result is b i1 a 1j + b i2 a 2j + + b in a nj, the desired result.

58 Counting Paths a b c d A = A 4 = Number of paths of length 4 from a to d is (1,4) th entry of A 4 = 8.

59 The Seven Bridges of Königsberg, Germany The residents of Königsberg, Germany, wondered if it was possible to take a walking tour of the town that crossed each of the seven bridges over the Presel river exactly once. Is it possible to start at some node and take a walk that uses each edge exactly once, and ends at the starting node?

60 The Seven Bridges of Königsberg, Germany You can redraw the original picture as long as for every edge between nodes i and j in the original you put an edge between nodes i and j in the redrawn version (and you put no other edges in the redrawn version). Original: 2 3 Redrawn:

61 The Seven Bridges of Königsberg, Germany Euler: Has no tour that uses each edge exactly once. (Even if we allow the walk to start and finish in different places.) Can you see why?

62 Euler - definitions An Eulerian path (Eulerian trail, Euler walk) in a graph is a path that uses each edge precisely once. If such a path exists, the graph is called traversable. An Eulerian cycle (Eulerian circuit, Euler tour) in a graph is a cycle that uses each edge precisely once. If such a cycle exists, the graph is called Eulerian (also unicursal). Representation example: G1 has Euler path a, c, d, e, b, d, a, b a b c d e

63 The problem in our language: Show that is not Eulerian. In fact, it contains no Euler trail.

64 Euler - theorems 1. A connected graph G is Eulerian if and only if G is connected and has no vertices of odd degree 2. A connected graph G is has an Euler trail from node a to some other node b if and only if G is connected and a b are the only two nodes of odd degree

65 Euler theorems (=>) Assume G has an Euler trail T from node a to node b (a and b not necessarily distinct). For every node besides a and b, T uses an edge to exit for each edge it uses to enter. Thus, the degree of the node is even. 1. If a = b, then a also has even degree. Euler circuit 2. If a b, then a and b both have odd degree. Euler path

66 Euler - theorems 1. A connected graph G is Eulerian if and only if G is connected and has no vertices of odd degree a b f c d e Building a simple path: {a,b}, {b,c}, {c,f}, {f,a} Euler circuit constructed if all edges are used. True here?

67 Euler - theorems 1. A connected graph G is Eulerian if and only if G is connected and has no vertices of odd degree c d e Delete the simple path: {a,b}, {b,c}, {c,f}, {f,a} C is the common vertex for this sub-graph with its parent.

68 Euler - theorems 1. A connected graph G is Eulerian if and only if G is connected and has no vertices of odd degree c d Constructed subgraph may not be connected. e C is the common vertex for this sub-graph with its parent. C has even degree. Start at c and take a walk: {c,d}, {d,e}, {e,c}

69 Euler - theorems 1. A connected graph G is Eulerian if and only if G is connected and has no vertices of odd degree a b f c e d Splice the circuits in the 2 graphs: {a,b}, {b,c}, {c,f}, {f,a} + {c,d}, {d,e}, {e,c} = {a,b}, {b,c}, {c,d}, {d,e}, {e,c}, {c,f} {f,a}

70 Euler Circuit 1. Circuit C := a circuit in G beginning at an arbitrary vertex v. 1. Add edges successively to form a path that returns to this vertex. 2. H := G above circuit C 3. While H has edges 1. Sub-circuit sc := a circuit that begins at a vertex in H that is also in C (e.g., vertex c ) 2. H := H sc (- all isolated vertices) 3. Circuit := circuit C spliced with sub-circuit sc 4. Circuit C has the Euler circuit.

71 Representation- Incidence Matrix e 1 e 2 e 3 e4 e 5 e 6 e 7 e1 a b a e7 e2 b f e6 e5 c e e3 e4 d c d e f

72 Homework 1 Write a program to obtain Euler Circuits. Input graphs can be Eulerian, no need for checking non Euler graphs Include a simple user interface to input the graph. Minimum of 10 edges (no more than 15 edges needed) Simple documentation Include a sample graph, if needed, to test Any programming language Submission on WebCT Due on January 27 th 11.55pm.

73 Hamiltonian Graph Hamiltonian path (also called traceable path) is a path that visits each vertex exactly once. A Hamiltonian cycle (also called Hamiltonian circuit, vertex tour or graph cycle) is a cycle that visits each vertex exactly once (except for the starting vertex, which is visited once at the start and once again at the end). A graph that contains a Hamiltonian path is called a traceable graph. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent.

74 A graph of the vertices of a dodecahedron. Is it Hamiltonian? Yes.

75 Hamiltonian Graph This one has a Hamiltonian path, but not a Hamiltonian tour.

76 Hamiltonian Graph This one has an Euler tour, but no Hamiltonian path.

77 Hamiltonian Graph Similar notions may be defined for directed graphs, where edges (arcs) of a path or a cycle are required to point in the same direction, i.e., connected tail-to-head. The Hamiltonian cycle problem or Hamiltonian circuit problem in graph theory is to find a Hamiltonian cycle in a given graph. The Hamiltonian path problem is to find a Hamiltonian path in a given graph. There is a simple relation between the two problems. The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G. Both problems are NP-complete. However, certain classes of graphs always contain Hamiltonian paths. For example, it is known that every tournament has an odd number of Hamiltonian paths.

78 Hamiltonian Graph DIRAC S Theorem: if G is a simple graph with n vertices with n 3 such that the degree of every vertex in G is at least n/2 then G has a Hamilton circuit. ORE S Theorem: if G is a simple graph with n vertices with n 3 such that deg (u) + deg (v) n fro every pair of nonadjacent vertices u and v in G, then G has a Hamilton circuit.

79 Shortest Path Generalize distance to weighted setting Digraph G = (V,E) with weight function W: E R (assigning real values to edges) Weight of path p = v 1 v 2 v k is k 1 wp ( ) = wv ( i, vi+ 1) i= 1 Shortest path = a path of the minimum weight Applications static/dynamic network routing robot motion planning map/route generation in traffic

80 Shortest-Path Problems Shortest-Path problems Single-source (single-destination). Find a shortest path from a given source (vertex s) to each of the vertices. The topic of this lecture. Single-pair. Given two vertices, find a shortest path between them. Solution to single-source problem solves this problem efficiently, too. All-pairs. Find shortest-paths for every pair of vertices. Dynamic programming algorithm. Unweighted shortest-paths BFS.

81 Optimal Substructure Theorem: subpaths of shortest paths are shortest paths Proof ( cut and paste ) if some subpath were not the shortest path, one could substitute the shorter subpath and create a shorter total path

82 Negative Weights and Cycles? Negative edges are OK, as long as there are no negative weight cycles (otherwise paths with arbitrary small lengths would be possible) Shortest-paths can have no cycles (otherwise we could improve them by removing cycles) Any shortest-path in graph G can be no longer than n 1 edges, where n is the number of vertices

83 Shortest Path Tree The result of the algorithms a shortest path tree. For each vertex v, it records a shortest path from the start vertex s to v. v.parent() gives a predecessor of v in this shortest path gives a shortest path length from s to v, which is recorded in v.d(). The same pseudo-code assumptions are used. Vertex ADT with operations: adjacent():vertexset d():int and setd(k:int) parent():vertex and setparent(p:vertex)

84 Relaxation For each vertex v in the graph, we maintain v.d(), the estimate of the shortest path from s, initialized to at the start Relaxing an edge (u,v) means testing whether we can improve the shortest path to v found so far by going through u u 5 u 2 2 v 9 Relax(u,v) 5 7 v u 5 u v 2 Relax (u,v,g) 6 Relax(u,v) v if v.d() > u.d()+g.w(u,v) then v.setd(u.d()+g.w(u,v)) v.setparent(u)

85 Dijkstra's Algorithm Non-negative edge weights Greedy, similar to Prim's algorithm for MST Like breadth-first search (if all weights = 1, one can simply use BFS) Use Q, a priority queue ADT keyed by v.d() (BFS used FIFO queue, here we use a PQ, which is re-organized whenever some d decreases) Basic idea maintain a set S of solved vertices at each step select "closest" vertex u, add it to S, and relax all edges from u

86 Dijkstra s ALgorithm Solution to Single-source (single-destination). Input: Graph G, start vertex s Dijkstra(G,s) 01 for each vertex u G.V() 02 u.setd( ) 03 u.setparent(nil) 04 s.setd(0) 05 S // Set S is used to explain the algorithm 06 Q.init(G.V()) // Q is a priority queue ADT 07 while not Q.isEmpty() 08 u Q.extractMin() 09 S S {u} 10 for each v u.adjacent() do 11 Relax(u, v, G) 12 Q.modifyKey(v) relaxing edges

87 Dijkstra s Example Dijkstra(G,s) 01 for each vertex u G.V() 02 u.setd( ) 03 u.setparent(nil) 04 s.setd(0) 05 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 u Q.extractMin() 09 S S {u} 10 for each v u.adjacent() do 11 Relax(u, v, G) 12 Q.modifyKey(v) s s u 2 3 x u v 4 6 y v x y

88 Dijkstra s Example Dijkstra(G,s) 01 for each vertex u G.V() 02 u.setd( ) 03 u.setparent(nil) 04 s.setd(0) 05 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 u Q.extractMin() 09 S S {u} 10 for each v u.adjacent() do 11 Relax(u, v, G) 12 Q.modifyKey(v) s s u v x u y v x y

89 Dijkstra s Example u v Dijkstra(G,s) 01 for each vertex u G.V() 02 u.setd( ) 03 u.setparent(nil) 04 s.setd(0) 05 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 u Q.extractMin() 09 S S {u} 10 for each v u.adjacent() do 11 Relax(u, v, G) 12 Q.modifyKey(v) x u y v x y

90 Dijkstra s Algorithm O(n 2 ) operations (n-1) iterations: 1 for each vertex added to the distinguished set S. (n-1) iterations: for each adjacent vertex of the one added to the distinguished set. Note: it is single source single destination algorithm

91 Traveling Salesman Problem Given a number of cities and the costs of traveling from one to the other, what is the cheapest roundtrip route that visits each city once and then returns to the starting city? An equivalent formulation in terms of graph theory is: Find the Hamiltonian cycle with the least weight in a weighted graph. It can be shown that the requirement of returning to the starting city does not change the computational complexity of the problem. A related problem is the (bottleneck TSP): Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge.

92 Planar Graphs A graph (or multigraph) G is called planar if G can be drawn in the plane with its edges intersecting only at vertices of G, such a drawing of G is called an embedding of G in the plane. Application Example: VLSI design (overlapping edges requires extra layers), Circuit design (cannot overlap wires on board) Representation examples: K1,K2,K3,K4 are planar, Kn for n>4 are non-planar K 4

93 Planar Graphs Representation examples: Q 3

94 Planar Graphs Representation examples: K 3,3 is Nonplanar v 1 v 2 v 3 v 1 v 5 v 1 v 5 R 2 R 1 R 21 R 22 R 1 v 3 v 4 v 5 v 6 v v v v 2

95 Planar Graphs Theorem : Euler's planar graph theorem For a connected planar graph or multigraph: v e + r = 2 number of vertices number of edges number of regions

96 Planar Graphs Example of Euler s theorem K 4 R2 R1 R4 A planar graph divides the plane into several regions (faces), one of them is the infinite region. R3 v=4,e=6,r=4, v-e+r=2

97 Planar Graphs Proof of Euler s formula: By Induction Base Case: for G1, e 1 = 1, v 1 = 2 and r 1 = 1 u R1 v n+1 Case: Assume, r n = e n v n + 2 is true. Let {an+1, bn+1} be the edge that is added to Gn to obtain Gn+1 and we prove that r n = e n v n + 2 is true. Can be proved using two cases.

98 Planar Graphs Case 1: r n+1 = r n + 1, e n+1 = e n + 1, v n+1 = v n => r n+1 = e n+1 v n a n+1 R b n+1

99 Planar Graphs Case 2: r n+1 = r n, e n+1 = e n + 1, v n+1 = v n + 1 => r n+1 = e n+1 v n a n+1 R b n+1

100 Planar Graphs Corollary 1: Let G = (V, E) be a connected simple planar graph with V = v, E = e > 2, and r regions. Then 3r 2e and e 3v 6 Proof: Since G is loop-free and is not a multigraph, the boundary of each region (including the infinite region) contains at least three edges. Hence, each region has degree 3. Degree of region: No. of edges on its boundary; 1 edge may occur twice on boundary -> contributes 2 to the region degree. Each edge occurs exactly twice: either in the same region or in 2 different regions a n+1 R b n+1

101 Region Degree R Degree of R = 3 R Degree of R =?

102 Planar Graphs Each edge occurs exactly twice: either in the same region or in 2 different regions 2e = sum of degree of r regions determined by 2e 2e 3r. (since each region has a degree of at least 3) r (2/3) e From Euler s theorem, 2 = v e + r 2 v e + 2e/3 2 v e/3 So 6 3v e or e 3v 6

103 Planar Graphs Corollary 2: Let G = (V, E) be a connected simple planar graph then G has a vertex degree that does not exceed 5 Proof: If G has one or two vertices the result is true If G has 3 or more vertices then by Corollary 1, e 3v 6 2e 6v 12 If the degree of every vertex were at least 6: by Handshaking theorem: 2e = Sum (deg(v)) 2e 6v. But this contradicts the inequality 2e 6v 12 There must be at least one vertex with degree no greater than 5

104 Planar Graphs Corollary 3: Let G = (V, E) be a connected simple planar graph with v vertices ( v 3), e edges, and no circuits of length 3 then e 2v -4 Proof: Similar to Corollary 1 except the fact that no circuits of length 3 imply that degree of region must be at least 4.

105 Planar Graphs Elementary sub-division: Operation in which a graph are obtained by removing an edge {u, v} and adding the vertex w and edges {u, w}, {w, v} u v u w v Homeomorphic Graphs: Graphs G1 and G2 are termed as homeomorphic if they are obtained by sequence of elementary sub-divisions.

106 Planar Graphs Kuwratoski s Theorem: A graph is non-planar if and only if it contains a subgraph homeomorephic to K 3,3 or K 5 Representation Example: G is Nonplanar a j k a b c d i f e g b a b c d c h e e g f d H K 5 G

107 Graph Coloring Problem Graph coloring is an assignment of "colors", almost always taken to be consecutive integers starting from 1 without loss of generality, to certain objects in a graph. Such objects can be vertices, edges, faces, or a mixture of the above. Application examples: scheduling, register allocation in a microprocessor, frequency assignment in mobile radios, and pattern matching

108 Vertex Coloring Problem Assignment of colors to the vertices of the graph such that proper coloring takes place (no two adjacent vertices are assigned the same color) Chromatic number: least number of colors needed to color the graph A graph that can be assigned a (proper) k-coloring is k-colorable, and it is k-chromatic if its chromatic number is exactly k.

109 C 4 C 5 K 4 K 2, 3 Vertex Coloring Problem The problem of finding a minimum coloring of a graph is NP-Hard The corresponding decision problem (Is there a coloring which uses at most k colors?) is NP-complete The chromatic number for C n = 3 (n is odd) or 2 (n is even), K n = n, K m,n = 2 Cn: cycle with n vertices; Kn: fully connected graph with n vertices; Km,n: complete bipartite graph

110 Vertex Covering Problem The Four color theorem: the chromatic number of a planar graph is no greater than 4 Example: G1 chromatic number = 3, G2 chromatic number = 4 (Most proofs rely on case by case analysis). G1 G2